Skip to main content
SpringerLink
Log in

Self-similar potentials and theq-oscillator algebra at roots of unity

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

Abstract

Properties of the simplest class of self-similar potentials are analyzed. Wave functions of the corresponding Schrödinger equation provide bases of representations of theq-deformed Heisenberg-Weyl algebra. When the parameterq is a root of unity, the functional form of the potentials can be found explicitly. The generalq 3 = 1 and the particularq 4 = 1 potentials are given by the equi-anharmonic and (pseudo) lemniscatic Weierstrass functions, respectively.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Infeld, L. and Hull, T. E.,Rev. Modern Phys. 23, 21 (1951).

    Google Scholar 

  2. Shabat, A.,Inverse Problems 8, 303 (1992).

    Google Scholar 

  3. Spiridonov, V.,Phys. Rev. Lett. 69, 398 (1992).

    Google Scholar 

  4. Spiridonov, V., inProc. XIXth ICGTMP, Salamanca, 29 June-4 July 1992, to appear; Preprint UdeM-LPN-TH-123, 1992, to appear inComm. Theor. Phys. (Allahabad).

  5. Kulish, P. P. and Reshetikhin, N. Yu.,Zap. Nauchn. Sem. LOMI 101, 101 (1981) (in Russian); Sklyanin, E. K.,Funct. Anal. Appl. 17, 273 (1983); Drinfel'd, V. G., inProc. ICM, Berkeley, 1986; Jimbo, M.,Lett. Math. Phys. 11, 247 (1986).

    Google Scholar 

  6. Biedenharn, L. C.,J. Phys. A22, L873 (1989); Macfarlane, A. J.,J. Phys. A22, 4581 (1989); Hayashi, T.,Comm. Math. Phys. 127, 129 (1990).

    Google Scholar 

  7. Exton, H.,q-Hypergeometric Functions and Applications, Ellis Horwood, Chichester, 1983; Atakishiyev, N. M. and Suslov, S. K., in A. A. Gonchar and E. B. Saff (eds.),Progress in Approximation Theory, Springer, New York, 1991.

    Google Scholar 

  8. Vaksman, L. L. and Soibel'man, Ya. S.,Funct. Anal. Appl. 22, 170 (1988); Masuda, T., Mimachi, K., Nakagami, Y., Noumi, M., Saburi, Y. and Ueno, K.,Lett. Math. Phys. 19, 187, 195 (1990); Floreanini, R. and Vinet, L.,Lett. Math. Phys. 22, 45 (1991); Kalnins, E. G., Manocha, H. L. and Miller, W.,J. Math. Phys. 33, 2365 (1992).

    Google Scholar 

  9. Roche, P. and Arnaudon, D.,Lett. Math. Phys. 17, 295 (1989); De Concini, C. and Kac, V. G.,Progr. Math. 92, 471 (1990).

    Google Scholar 

  10. Shabat, A. B. and Yamilov, R. I.,Leningrad Math. J. 2, 377 (1991).

    Google Scholar 

  11. Its, A. R. and Matveev, V. B.,Soviet J. Theoret. Math. Phys. 23, 343 (1975).

    Google Scholar 

  12. Shabat, A. B. and Veselov, A. P., Preprint of Forschungsinstitut für Mathematik ETH Zürich, 1992, to appear inFunct. Anal. Appl.

  13. El'sgol'ts, L. E. and Norkin, S. B.,Introduction to the Theory and Application of Differential Equations with Deviating Arguments, Academic Press, New York, 1973; Kamenskii, G. A.,Mat. Sb. 55(97), 363 (1961).

    Google Scholar 

  14. Novokshenov, V. Yu., Clarkson University preprint, INS-203, 1992.

  15. Abramowitz, M. and Stegun, I. A.,Handbook of Mathematical Functions, Nat. Bur. of Standards, 1966.

  16. Olshanetsky, M. A. and Perelomov, A. M.,Phys. Rep. 94, 322 (1983).

    Google Scholar 

Download references

Author information

Author notes

  1. Sergei Skorik

    Present address: Moscow Institute of Physics and Technology, Moscow, Russia

  2. Vyacheslav Spiridonov (International NSERC Fellow)

    Present address: Institute for Nuclear Research, Moscow, Russia

Authors and Affiliations

  1. Department of Physics, University of Southern California, 90089-0484, Los Angeles, CA, USA

    Sergei Skorik

  2. Laboratoire de Physique Nucléaire, Université de Montréal, CP 6128, succ. A, H3C 3J7, Montréal, Québec, Canada

    Vyacheslav Spiridonov (International NSERC Fellow)

Authors
  1. Sergei Skorik
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vyacheslav Spiridonov
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Skorik, S., Spiridonov, V. Self-similar potentials and theq-oscillator algebra at roots of unity. Lett Math Phys 28, 59–74 (1993). https://doi.org/10.1007/BF00739567

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00739567

Mathematics Subject Classifications (1991)

Search

Navigation